PREDATOR-PREY MODEL WITH CONSTANT RATE OF … · dengan hasil kepadatan sesuatu populasi. Populasi...

PREDATOR-PREY MODEL WITH CONSTANT RATE OF HARVESTING

NURUL AINA BINTI ARIFIN

A thesis submitted in fulfillment of the

requirements for the awards of the degree of

Master of Science (Mathematics)

Faculty of Science

Universiti Teknologi Malaysia

JANUARY 2014

iii

Special dedicate to my beloved family

Arifin Bin Abdullah

Rohani Binti Dali

Mohd Fakharuddin Bin Arifin

Nurulain Hanis Binti Shahruni

Mohamad Amirudin Bin Arifin

and

Nurul Atiqah Arifin

Thanks for all the supports, sacrifices, patients and willingness to share my dreams

with me...

To my honoured supervisor, Dr Faridah Mustapha, my friends (Mira, Hidayah,

Yana, Azira, Amiza and Aisya) and all UTM lecturers

Thank you for everything you taught me. Giving me inspiration to be knowledge

person.

iv

ACKNOWLEDGMENT

All praise belongs to Allah (SWT), The Lord of the Universe, without the

health, strength and perseverance he gave, I would not be able to complete this

thesis.

First and foremost, I would like to express my gratefulness and appreciation

to my supervisor Dr. Faridah Mustapha for her guidance and support that she had

given me along the completion of this project.

I also feel grateful to PSZ for providing me the information for my research

study.

Finally, I also would like to thank all my beloved persons in my life that

inspire and motivate me a lot throughout all the hard works while carrying out this

research, especially to my beloved parents, Arifin Abdullah and Rohani Dali, my

brother, Fakharudin, my little brother and sister, Amirudin and Atiqah. Last but not

least, thanks to all my friends for sharing ideas and giving encouragement.

v

ABSTRACT

Predator-prey model is the first model to illustrate the interaction between

predators and prey. It is a topic of great interest for many ecologists and

mathematicians. This model assumes that the predator populations have negative

effects on the prey populations. The generalized equation of this model is the

response of the populations would be proportional to the product of their population

densities. Prey population grows with limited by carrying capacity, K and it is called

the logistic equation. Thus, in this research, there are four different cases are

analyzed which are predator-prey model, predator-prey model with harvesting in

prey, predator-prey model with harvesting in predator and predator-prey model with

harvesting in both populations. Systems of ordinary differential equation are used for

all models. The objectives of this research are i) to study the concept of Lotka-

Volterra predator-prey model, ii) to analyze the predator-prey model with constant

rate of harvesting in prey, iii) to analyze the predator-prey model with constant rate

of harvesting in predator, iv) to analyze the predator-prey model with constant rate of

harvesting in both populations. In analyzing all four models, equilibrium points will

be obtained and analyzed for the stability by using Routh-Hurwitz Criteria. Lastly,

some numerical examples and graphical analysis are shown to illustrate the stability

of the stable equilibrium points and the effects of harvesting to the systems.

vi

ABSTRAK

Model Pemangsa-Mangsa merupakan model yang pertama untuk

menggambarkan interaksi antara pemangsa dan mangsa. Ia adalah satu topik yang

menarik minat besar bagi semua ahli ekologi dan ahli matematik. Model ini

mengandaikan populasi pemangsa memberi kesan negatif kepada populasi mangsa.

Persamaan umum model ini adalah tindak balas sesuatu populasi yang berkadar

dengan hasil kepadatan sesuatu populasi. Populasi mangsa membesar dengan

dihadkan oleh keupayaan membawa, K dan ia dipanggil sebagai persamaan logistik.

Oleh itu, dalam kajian ini, terdapat empat kes berbeza yang dianalisis iaitu model

pemangsa-mangsa, model pemangsa-mangsa dengan penangkapan dalam mangsa,

model pemangsa-mangsa dengan penangkapan dalam pemangsa dan model

pemangsa-mangsa dengan penangkapan dalam kedua-dua populasi. Sistem

persamaan pembezaan biasa digunakan untuk keseluruhan model. Objektif kajian ini

adalah i) untuk mengkaji konsep model Lotka-Volterra Pemangsa-mangsa, ii) untuk

menganalisis pemangsa-mangsa model dengan kadar tetap penangkapan dalam

mangsa, iii) untuk menganalisis pemangsa-mangsa model dengan kadar tetap

penangkapan dalam pemangsa, iv) untuk menganalisis pemangsa-mangsa model

dengan kadar tetap penangkapan dalam kedua-dua populasi. Dalam menganalisis

keempat-empat model tersebut, titik keseimbangan akan diperolehi dan

kestabilannya akan dianalisis dengan menggunakan Kriteria Routh-Hurwitz. Akhir

sekali, beberapa contoh berangka dan analisis graf ditunjukkan untuk

memperlihatkan kestabilan titik keseimbangan dan kesan penangkapan dalam

sistem.

vii

TABLE OF CONTENTS

CHAPTER TITLE PAGE

DECLARATION

DEDICATION

ACKNOWLEDGEMENT

ABSTRACT

ABSTRAK

TABLE OF CONTENTS

LIST OF SYMBOLS

LIST OF FIGURES

LIST OF TABLES

LIST OF APPENDICES

ii

iii

iv

v

vi

vii

xi

xii

xv

xvi

1 INTRODUCTION

1.1 Background of Research

1.2 Problem Statement

1.3 Objectives of Research

1.4 Scope of Research

1.5 Significant of Research

1

3

3

4

4

2 LITERATURE REVIEW

2.1 Introduction

2.2 Mathematical Modeling

5

6

viii

2.3 Predator-Prey Interaction

2.4 Lotka-Volterra Predator-Prey Model

2.5 Predator-Prey Model with Constant Rate of

Harvesting

2.5.1 Definition of Harvesting in Population

Dynamics

2.5.2 Stability with Harvesting

7

8

9

10

11

3 RESEARCH METHODOLOGY

3.1 Introduction

3.2 Nullclines

3.3 Equilibrium Analysis

3.4 Eigenvalues

3.4.1 Asymptotically Stability

3.5 Routh – Hurwitz Criteria

3.6 Numerical Method to Solve Ordinary Differential

Equation

14

15

16

17

19

20

21

4 PREDATOR-PREY MODEL

4.1 Introduction

4.2 The Model

4.3 Equilibrium Points of Predator-Prey Model

4.4 Analyzing the Model

4.5 Numerical Examples

23

24

26

28

33

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5 PREDATOR-PREY MODEL WITH CONSTANT

RATE OF PREY HARVESTING

5.1 Introduction

5.2 The Model

5.3 Equilibrium Points of Predator-Prey Model with

Harvesting in Prey



38

39

42

45

51


RATE OF PREDATOR HARVESTING

6.1 Introduction

6.2 The Model


Harvesting in Predator



58

59

62

65

69


RATE OF BOTH POPULATION HARVESTING

7.1 Introduction

7.2 The Model


Harvesting in Both Populations



73

74

77

79

83

x

8 CONCLUSION AND RECOMMENDATIONS

8.1 Introduction

8.2 Research Conclusion

8.3 Recommendations

87

88

90

REFERENCES 91

APPENDICES 94

xi

LIST OF SYMBOLS

( )x t - Prey population

( )y t - Predator population

xH - Prey harvesting

yH - Predator harvesting

1,2 - Eigenvalues of equilibrium point

r - Growth rate of prey

K - Carrying capacity of the prey.

- Rate of consumption of prey by predator.

- Conversion of prey consumed into predator reproduction rate.

c - Death rate of predator

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LIST OF FIGURES

FIGURE

NO. TITLE PAGE

Figure 3.1 Phase portraits for real or distinct eigenvalues 18

Figure 3.2 Phase portraits for complex eigenvalues 19

Figure 4.1 The death rate of predator, K versus the conversion of prey

into predator reproduction rate, c with fixed values for 1r

and 1 .

32

Figure 4.2 Phase portraits for a nonlinear system (4.10) 34

Figure 4.3 The graph show population densities of prey and predator

versus time, 1, 1, 2, 0.05 and 0.3r K c

35


Figure 4.5 The graph show population densities of prey and predator

versus time, 1, 1, 10, 0.05 and 0.3r K c

37

Figure 5.1 Constant rate of prey harvesting, xH versus growth rate of prey

and carrying capacity, rK

43

Figure 5.2 Constant rate of prey harvesting, xH versus death rate of

predator, c with 1, 2, 1 and 0.05r K .

50


xiii

Figure 5.4 Phase portraits for a nonlinear system with 0.07c and

0.45xH

53

Figure 5.5 The graphs show the population densities of prey and predator

versus time, 1, 2, 1, 0.05r K . a) 0.09, 0.4xc H

b) 0.07, 0.45xc H

53


Figure 5.7 Phase portraits for a nonlinear system with 0.07 andc

0.3xH

56

Figure 5.8 The graphs show the population densities of prey and predator

versus time, 1, 2, 1, 0.05r K . a) 0.08, 0.2xc H

b) 0.07, 0.3xc H

56

Figure 6.1 Constant rate of predator harvesting, yH versus growth rate of

prey, r with 10, 1, 0.05 and 0.3K c

64

Figure 6.2 Phase portraits of nonlinear system (6.20) with 1 andr

0.01yH

70

Figure 6.3 Phase portraits with 2 and 0.02yr H 71

Figure 6.4 The graph show the population densities of prey and predator

versus time 10, 1, 0.05, 0.3K c

a) 1 and 0.01yr H b) 2 and 0.02yr H

71

Figure 7.1 Constant rate of prey harvesting, xH versus growth rate of prey

and carrying capacity, rK

1, 0.05, 0.3 and 0.02yc H

78

xiv

Figure 7.2 Constant rate of prey harvesting, xH versus the conversion of

prey consumed into predator reproduction rate, with

( 1, 20, 1, 0.3 and 0.02yr K c H ).

85

Figure 7.3 The graph show the population densities of prey and predator

versus time, 1, 100, 1, 0.05, 0.3, 0.01 xr K c H

and 0.02yH

86

xv

LIST OF TABLES

TABLE

NO. TITLE PAGE

5.1 Parameters used in predator-prey model with constant

rate of harvesting in prey

41

8.1 Summarization on the stability of equilibrium points with

the conditions for predator-prey model without

harvesting, predator-prey model with harvesting in prey,

predator-prey model with harvesting in both populations.

89

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LIST OF APPENDICES

APPENDIX TITLE PAGE

A Calculation of Eigenvalues for Each Equilibrium Points

and Calculation of Fourth Order Runge Kutta Using Maple

Software.

94

B Phase Portraits of the Predator-Prey Model with Constant

Rate of Prey Harvesting using Matlab Software.

96

CHAPTER 1

INTRODUCTION

1.1 Background of Research

In the ecology system, the predator-prey model is amongst the oldest studies

and also the first model to illustrate the interaction between predators and prey. This

model assume that the predator populations have negative effects on the prey

populations and this system was formulated by Vito Volterra, an Italian

mathematician and Alfred Lotka, an American mathematical biologist in 1925

(Boyce and DiPrima, 2010).

There are several known predator-prey models such as Lotka-Volterra model,

Logistic equation, Holling Tanner Type 2 and Type 3 models, Yodzis model and so

on. This study only focuses on the most well known Lotka-Volterra model. This

model has been analyzed by various text books in dynamical systems, mathematical

biology, ecology, differential equations etc. The generalized equation of this model is

the response of the populations would be proportional to the product of their

population densities. The prey population grows infinitely in the absence of

predators. Therefore, the logistic equation or often called as carrying capacity of the

environment was added to the prey equation given,

2

( )( ) 1 ( ) ( ),

( ) ( ) ( ),

dx x trx t x t y t

dt K

dyx t y t cy t

dt

(1.1)

where

( )x t denotes prey population.

( )y t denotes predator population.

r denotes the growth rate of prey.

K denotes carrying capacity.

denotes the rate of consumption of prey by predator.

denotes the conversion of prey consumed into predator reproduction

rate.

c denotes death rate of the predator.

Our study also continues by adding constant rate of harvesting to the model.

In the previous research, Brauer (1977), Dai and Tang (1998), Martin and Ruan

(2001), Kar and Pahari (2006), Syamsuddin and Malik (2008), Xia et al. (2009) and

Agarwal and Pathak (2012) had analyzed the constant rate of harvesting either prey

or predator population. According to Kar and Pahari (2006), many of interesting

dynamical behaviors such as the stability of the equilibrium points, existence of Hopf

bifurcation and limit cycles have been observed. Therefore, this study concern on

analyze the stability of the model by adding the harvesting to both populations by

using the method of differential equation.

Lotka-Volterra predator-prey model plays a crucial role in the population

dynamics and also one of the major advanced theories introduced by Lotka and

Volterra. They assumed the response of the population would be proportional to the

product of their population densities without any delayed and constant rate of

harvesting. The model formulation is in the equation (1.1). Some of the studies found

that the rate of harvesting has been used to control the increasing of population and

as a controller of the population density (Ouncharoen et al., 2010). This means that

3

harvesting can make whether the population increases or decreases for a continuity

yields but it will limited by a carrying capacity in equation (1.1).

In this project, the constant rate of harvesting will be added to either preys,

predators or both populations and we will analyze the stability of the equilibrium

points of the predator-prey model.

1.2 Problem Statement

This study is focused on how constant rate of harvesting affect the dynamics

of the predator-prey system.

1.3 Objectives of Research

The objectives of this research are:

1. To study the concept of Lotka-Volterra predator-prey model.

2. To analyze the predator-prey model with constant rate of harvesting in

prey.


predator.


both populations.

4

1.4 Scope of Research

The main scope of this research is to analyze the predator-prey model by with

the constant rate of harvesting. In this research, we shall only focus on two

populations which are prey and predator. We will formulate the model by adding the

constant rate of harvesting prey, predator or both population and also find how this

factor affects to the stability of equilibrium points in predator-prey model.

1.5 Significant of Research

The findings of this research is useful for the mathematicians who are

interested in the ecology fields because this research will give us more

understanding in predator-prey model with or without harvesting and the effects on

the stability of the population. The result obtained can be a guide by applying to

another predation model in population field.

91

REFERENCES

Agarwal, M. and Pathak, R. (2012). Harvesting and Hopf Bifurcation in a prey-

predator model with Holling Type IV Functional Response. International

Journal of Mathematics and Soft Computing. Vol.2, No.1 (2012), 83 – 92.

Barnes, B. and Fulford, G. R. (2009). Mathematical Modelling with Case Studies: A

Differential Approach Using Maple nad Matlab. 2nd edition. New York:

CRC Press, Taylor & Francis Group, A Chapman & Hall Book .

Boyce, W. E. and DiPrima, R. C. (2010). Elementary Differential Equations and

Boundary Value Problems. 9th edition. New York: John Wiley & Sons

(Asia) Pte Ltd.

Brauer ,F. (1977). Stability of some population models with delay. Math. Biosci.,

33, 345-358.

Brauer, F. and Soudack, A. C. (1979). Stability Regions and transition phenomena

for harvested predator–prey systems. Journal of Mathematical Biology 7,

319–337.

Burghes, D., Galbraith, P., Price, N. and Sherlock, A. (1996). Mathematical

Modeling. UK: Prentice Hall.

Dai, G and Tang, M. (1998). Coexistence region and global dynamics of a harvested

predator-prey system. Siam Journal of Applied Mathematics (1998), 58,

193-210.

Farlow, J., Hall, J. E., Mc Dill, J. E, and West, B. H. (2007). Differential Equations

with Dynamical System. New Jersey: Princeton University Press.

Giardano, F. R., Weir, M. D. and Fox, W. P. (2003). A First Course in Mathematical

Modeling. 3rd edition. US: Thomson BRooks/Cole.

92

Hennig, C.(2009). Mathematical Models and Reality- a Constructivist Perspective.

Research Report No. 304, Department of Statistical Science, University

College London.

Jessie, S. (2004). Population Dynamics Beyond Classic Lotka-Volterra Models.

Thesis: Degree of Bachelor of Arts in Liberal Arts and Sciences with a

Concentration in Mathematics. Harriet L. Wilkes Honors College of Florida

Atlantic University

Jones, D. S., Plank, M, J. and Sleeman, B, D. (2010). Differential Equations and

Mathematical Biology. Second Edition. UK: CRC Press Taylor & Francis

Group.

Kar, T. K. and Pahari, U. K. (2006). Non-selective harvesting in prey–predator

models with delay. Communications in Nonlinear Science and Numerical

Simulation 11 (2006) 499–509.

Kar, T. K. and Chakraborty, K. (2010). Effort Dynamics in a Prey-Predator Model

with Harvesting. International Journal Of Information And Systems

Sciences Volume 6, Number 3, Pages 318-332.

Keshet, L. E. (1988). Mathematical Models in Biology. New York: Random House,

Inc.

Mahony, J. J. and Fowkes, N. D. (1994). An Introduction to Mathematical

Modeling. New York: John Wiley & Sons, Inc.

Martin, A. and Ruan, S. (2001). Predator-prey models with delay and prey

harvesting. J. Mathematical Biology, 43 (2001), 247-267.

May, R. M. (1972). Limit cycles in predator–prey communities. Science 177, 900–

902.

Murray, J. D. (2002). Mathematical Biology 1. An Introduction. Third edition. US:

Springer Science+Business Media, Inc.

Narayan, K. L. and Ramacharyulu, N.C.P. (2008). A prey-predator model with an

alternative food for the predator, harvesting of both the species and with a

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gestation period for interaction. Int. J. Opne Problems Compt. Math., 1,

No. 1.

Ouncharoen R., Pinjai, S., Dumrongpokaphan, Th. and Lenbury, Y. (2010). Global

Stability Analysis of Predator-Prey Model with Harvesting and Delay. Thai

Journal of Mathematics Volume 8 (2010) Number 3: 589–605.

Salleh, K.. (2013). Dynamics of Modified Leslie-Gower Predator-Prey Model with

Predator Harvesting. International Journal of Basic & Applied Sciences IJBAS-

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“Harvesting a Prey Population”,

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